Question: Flannery used $30$ lilies and $78$ roses to create six identical flower arrangements. Write an equation to describe the relationship between $l$, the number of lilies, and $r$, the number of roses.
Explanation: Let's find the constant of proportionality. In the proportional relationship between $l$, the number of lilies, and $r$, the number of roses, one constant of proportionality is the number of roses per lily. It is also the number we multiply by the number of lilies to get the number of roses. $l\,\times\, ?=r$ $\begin{aligned} l\,\times\, {?}&=r \\\\ {?}&=\dfrac{r}{l} \\\\ &=\dfrac{78}{30} \\\\ &=\dfrac{13\times\cancel{6}}{5\times\cancel{6}} \\\\ &={\dfrac{13}{5}} \end{aligned}$ The constant of proportionality is ${\dfrac{13}{5}}$. This means we can multiply ${\dfrac{13}{5}}$ by the number of lilies to get the number of roses. Now, let's write the equation: $\begin{aligned} \text{number of roses}&={\text{number of roses per lily}}\times\text{number of lilies} \\\\ r&={\dfrac{13}{5}}l \end{aligned}$ One correct equation is: $r=\dfrac{13}{5}l$